Evaluate the determinant \( \det \begin{pmatrix} 2 & 1 & 3 \\ 1 & 0

Evaluate the determinant \( \det \begin{pmatrix} 2 & 1 & 3 \\ 1 & 0 & 2 \\ 3 & 4 & 1 \end{pmatrix} \).

Evaluate the determinant \( \det \begin{pmatrix} 2 & 1 & 3 \\ 1 & 0 & 2 \\ 3 & 4 & 1 \end{pmatrix} \).

Step 1: Write down the matrix: \( A = \begin{pmatrix} 2 & 1 & 3 \ 1 & 0 & 2 \ 3 & 4 & 1 \ \end{pmatrix} \).
Step 2: Identify the elements of the matrix: \( a = 2, b = 1, c = 3, d = 1, e = 0, f = 2, g = 3, h = 4, i = 1 \).
Step 3: Use the formula for the determinant of a 3x3 matrix: \( \det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \).
Step 4: Calculate each part of the formula:
- Calculate \( ei - fh = 0*1 - 2*4 = 0 - 8 = -8 \).
- Calculate \( di - fg = 1*1 - 2*3 = 1 - 6 = -5 \).
- Calculate \( dh - eg = 1*4 - 0*3 = 4 - 0 = 4 \).
Step 5: Substitute these values back into the determinant formula: \( \det(A) = 2(-8) - 1(-5) + 3(4) \).
Step 6: Simplify each term:
- First term: \( 2(-8) = -16 \).
- Second term: \( -1(-5) = 5 \).
- Third term: \( 3(4) = 12 \).
Step 7: Add these results together: \( -16 + 5 + 12 = -16 + 17 = 1 \).
Step 8: The final result is: \( \det(A) = -10 \).

Determinant Calculation – The process of calculating the determinant of a 3x3 matrix using the formula involving minors and cofactors.
Matrix Operations – Understanding how to perform basic operations on matrices, including multiplication and addition, which are often involved in determinant calculations.